Reconciling Hayek’s and Keynes’ Views of Recessions

Recessions often happen after periods of rapid accumulation of houses, consumer durables and business capital. This observation has led some economists, most notably Friedrich Hayek, to conclude that recessions often reﬂect periods of needed liquidation resulting from past over-investment. According to the main proponents of this view, government spending or any other form of aggregate demand policy should not be used to mitigate such a liquidation process, as doing so would simply result in a needed adjustment being postponed. In contrast, ever since the work of Keynes, many economists have viewed recessions as periods of deﬁcient demand that should be countered by activist ﬁscal policy. In this paper we reexamine the liquidation perspective of recessions in a setup where prices are ﬂexible but where not all trades are coordinated by centralized markets. The model illustrates why liquidations likely cause recessions characterized by deﬁcient aggregate demand and accordingly suggests that Keynes’ and Hayek’s views of recessions may be closely linked. In our framework, interventions aimed at stimulating aggregate demand face a trade-oﬀ whereby current stimulus postpones the adjustment process and therefore prolongs the recessions, but where some stimulative policies may nevertheless remain desirable.


INTRODUCTION
There remains considerable debate regarding the causes and consequences of recessions. Two views that are often presented as opposing, and which created controversy in the recent recession and its aftermath, are those associated with the ideas of Hayek and Keynes. 1 The Hayekian perspective is generally associated with viewing recessions as a necessary evil. According to this view, recessions mainly reflect periods of liquidation resulting from past over-accumulation of capital goods. A situation where the economy needs to liquidate such an excess can quite naturally give rise to a recession, but government spending aimed at stimulating activity, it is argued, is not warranted since it would mainly delay the needed adjustment process and thereby postpone the recovery. In contrast, the Keynesian view suggests that recessions reflect periods of deficient aggregate demand where the economy is not effectively exploiting the gains from trade between individuals. According to this view, policy interventions aimed at increasing 1. In response to the large recession in the US and abroad in 2008-2009, a high-profile debate around these two views was organized by Reuters (see http://www.reuters.com/subjects/keynes-hayek). See also Wapshott (2012) for a popular account of the Hayek-Keynes controversy. investment and consumption are generally desirable, as they favor the resumption of mutually beneficial trade between individuals. 2 In this paper we reexamine the liquidationist perspective of recessions in an environment with decentralized markets, flexible prices and search frictions. In particular, we examine how the economy adjusts when it inherits from the past an excessive amount of capital goods, which could be in the form of houses, durable goods or productive capital. Our goal is not to offer a novel explanation of why the economy may have overaccumulated in the past, 3 but to ask how it reacts to such an over-accumulation once it is realized. As suggested by Hayek, such a situation can readily lead to a recession as less economic activity is generally warranted when agents want to deplete past overaccumulation. However, because of the endogenous emergence of unemployment risk in our set-up, the size and duration of the recession implied by the need for liquidation is not socially optimal. In effect, the reduced gains from trade between individuals induced by the need for liquidation creates a multiplier process that leads to an excessive reduction in activity. Although prices are free to adjust, the liquidation creates a period of deficient aggregate demand where economic activity is too low because people spend too cautiously due to increased unemployment risk. In this sense, we argue that liquidation and deficient aggregate demand should not be viewed as alternative theories of recessions but instead should be seen as complements, where past over-accumulation may be a key driver of periods of deficient aggregate demand. This perspective also makes salient the trade-offs faced by policy. In particular, a policy-maker in our environment faces an unpleasant trade-off between the prescriptions emphasized by Keynes and Hayek. On the one hand, a policy-maker would want to stimulate economic activity during a liquidation-induced recession because consumers are too cautious. On the other hand, the policy-maker also needs to recognize that intervention will likely postpone recovery, since it slows down the needed depletion of excess capital. The model offers a simple framework where both of these forces are present and can be compared.
One potential criticism of a pure liquidationist view of recessions is that, if markets functioned efficiently, such periods should not be very socially painful. In particular, if economic agents interact in perfect markets and realize they have over-accumulated in the past, this should lead them to enjoy a sort of holiday paid for by their past excessive work. Looking backwards in such a situation, agents may resent the whole episode, but looking forward after a period of over-accumulation, they should nonetheless feel content to enjoy the proceeds of the past excessive work, even if it is associated with a recession. In contrast, in our environment we will show that liquidation periods are generally socially painful because of the multiplier process induced by cautious spending and unemployment risk. In effect, we will show that everyone in our model economy can be worse off when they inherit too many capital goods from the past. This result, whereby abundance creates scarcity, may appear quite counter-intuitive at first pass. To make as clear as possible the mechanism that can cause welfare to be reduced by such abundance, much of our analysis will focus on the case where the inherited capital takes the form of a good that directly contributes to utility, such as houses or durable goods. In this situation we 2. See Caballero and Hammour (2005) for an alternative view on the inefficiency of liquidations, based on the reduction of cumulative reallocation and inefficient restructuring in recessions.
3. There are several reason why an economy may over-accumulate capital. For example, agents may have had overly optimistic expectations about future expected economic growth that did not materialize, as in Beaudry and Portier (2004), or it could have been the case that credit supply was unduly subsidized either through explicit policy, as argued in  and , or as a by-product of monetary policy, as studied by Bordo and Landon-Lane (2013).
will show why inheriting too many houses or durables can make everyone worse off when decisions are decentralized.
A second potential criticism of a pure liquidationist view of recessions is that it often fails to explain why the economy does not simply reallocate factors to nondurables-producing sectors during the liquidationist period, and thereby maintain high employment. This criticism of the liquidationist view has been made forcefully, among others, by Krugman (1998). In particular, this line of criticism argues that since recessions are generally characterized by decreased production in almost all sectors, this constitutes clear evidence against the liquidationist view. In this paper we show why the coordination problem that arises initially in the durable goods sector as a result of past overaccumulation can create a contagion effect that causes consumption of both durable and non-durable goods to decrease simultaneously. The force that links the markets, and that makes them function as complements instead of substitutes, is precautionary behavior. Once there is less demand in the durable goods sector, agents fearing unemployment reduce demand in both sectors, thereby causing the decreased demand in the durable sector to spillover into non-durables.
The structure of our model builds on the literature related to search models of decentralized trading. In particular, we build on Lucas (1990), Shi (1998), Lagos and Wright (2005) and Rocheteau and Wright (2005) by allowing alternating decentralized and centralized markets to allow for a simple characterization of the equilibrium. However, unlike those papers, we do not have money in our setup. The paper also shares key features with the long tradition of macro models emphasizing strategic complementarities, aggregate demand externalities and multipliers, such as Diamond (1982), Cooper and John (1988) and more recently Angeletos and La'O (2013).
Unemployment risk and its effects on consumption decisions is at the core of our model. The empirical relevance of precautionary saving related to unemployment risk has been documented by many, including Carroll (1992), Carroll and Dunn (1997), Carroll et al. (2012) and Alan et al. (2012). There are also recent theoretical papers that emphasize how unemployment risk and precautionary savings can amplify shocks and cause business cycle fluctuations. These papers are the closest to our work. For example, Challe and Ragot (2013) have proposed a tractable quantitative model in which uninsurable unemployment risk is the source of wealth heterogeneity. Our model structure is probably most closely related to that presented in Guerrieri and Lorenzoni (2009). However, their model emphasizes why the economy may exhibit excessive responses to productivity shocks, while our framework offers a mechanism that amplifies demandtype shocks. Our paper also shares many features with Heathcote and Perri (2012), who develop a model in which unemployment risk and wealth impact consumption decisions. They focus on a strong form of demand externality that gives rise to multiple equilibria. 4 Finally, the works by Ravn and Sterk (2012), den Haan et al. (2014) and Challe et al. (2015) emphasize, as we do, how unemployment risk and precautionary savings can amplify demand shocks, but the mechanisms in these papers differs substantially from ours since they rely on nominal rigidities and constrained monetary policy. While the main mechanism in our model has many precursors in the literature, we believe that our setup illustrates most clearly (i ) how unemployment risk gives rise to a multiplier process for demand shocks even in the absence of price stickiness or increasing returns, (ii ) how this multiplier process can be ignited by periods of liquidation, and (iii ) how policy can and cannot be used to counter the process.
The remaining sections of the paper are structured as follows. In Section 2, we present a few motivating facts regarding the liquidationist view of recessions. In Section 3, we present a static model where agents inherit from the past a given level of capital goods, and we describe how and why high values of inherited capital can lead to deficient demand and poor economic outcomes. We begin the analysis with a model featuring only durable goods. We then extend it to the case of both durable and non-durable goods and show why inheriting many durable goods can also cause a reduction in non-durable purchases even when preferences are separable between the two types of goods. Finally, we extend the static model to the case where the stock of durable goods/houses can be used as collateral when financial markets are imperfect. In Section 4, we extend the model to an infinite-period dynamic setting and emphasize how the economy's behavior is different when it is close to rather than far from the steady state. We then calibrate the model to illustrate how it can provide an explanation of the observations we presented in Section 2. Finally, in Section 5, we discuss the trade-offs faced by a policy-maker in our dynamic setup. Section 6 concludes.

MOTIVATING OBSERVATIONS
The liquidationist view of recessions suggests that the severity and depth of a recession should reflect the extent to which an economy was in a situation of over-accumulated capital prior to the downturn. If the economy had gone through a period of substantial over-accumulation in prior years, then according to the liquidationist view the subsequent recession should be particularly deep and prolonged, as there would be a need to deplete a large amount of capital. In this section, we want to present some suggestive evidence in support of this idea. We first look at a comprehensive measure of capital, "total capital ", which consists of fixed capital plus durable goods. In the left panel of Figure 1 we present a scatter plot of the relationship between the depth of postwar US recessions 5 and a measure of prior over-investment. To create this measure of capital over-accumulation, we first construct a series of cumulated investment by summing past investments using the perpetual-inventory method over a rolling period of 40 quarters, and then remove a cubic time trend intended to capture the secular changes in growth rates over the period. 6 Our measure of the depth of a recession is the percentage fall in per capita real GDP from the preceding peak to the trough of the recession. Further details about the data and the construction of our measures are presented in Appendix A.

Figure 1
As can be seen in Figure 1, there is a very strong positive correlation between our measure of capital over-accumulation prior to a recession and the subsequent severity of the recession. While this evidence is only suggestive, it does support the notion that severe recessions have generally been preceded by periods of very high investment relative to 5. Specifically, our sample runs from 1957Q1 to 2015Q2. 6. Results are robust to using different low-frequency trend-removal methods, including an HP filter. However, we believe using a low-order polynomial trend to remove low frequency movements in growth rates is more conservative, since the resulting estimated trend it virtually not affected by recessions. In fact, the cubic trend we estimate is very close to linear, with a slight slow down in the 70s and 80s. secular growth. When we correlate our measure of capital accumulation with the length of the recovery -i.e., the time it takes for per capita real GDP to reach its previous peak level -we also find a strong positive and significant relationship, as shown in the right panel of Figure 1. It is worth pointing out that these correlations remain strongly positive and significant even if we exclude the most recent recession. 7 An interesting aspect of these results is that severe recessions were generally preceded by high accumulation of all three classes of capital that constitute total capital. In Table  1, we report the results of the two following regressions for various measures of capital: (2.1) where n indexes recessions, x is either the depth of the recession or the length of the recovery, ci represents deviations from trend of accumulated investment for either total, durable, non-residential or residential capital and ∆tf p n is the percentage change in TFP from peak to trough. As shown in Table 1, a positive correlation between our measure of capital-abundance and the subsequent depth and length of recessions holds for each of the four individual types of capital goods. Furthermore, note that this pattern holds even when the change in TFP from peak to trough is included as a control variable (i.e., equation (2.2)), which implies that the observed pattern is not confounded with effects potentially associated with technological change. In fact, we find that the change in TFP during recessions always enters insignificantly; that is, deeper and longer recessions are not observed to be associated with larger concurrent drops in TFP. 8 Table 1 Overall, we take this evidence as suggestive of a link between prior accumulation of capital and the severity of a recession, in line with the Hayekian view. It is worth noting that this does not necessarily imply that recessions are efficient adjustments of the economy, as our model will shortly illustrate.

STATIC MODEL
In this section, we present a stripped-down static model in order to illustrate as simply as possible why an economy may function particularly inefficiently when it inherits a large stock of capital from the past. In particular, we want to make clear why agents in an economy can be worse off when the stock of inherited capital goods is too high. For the mechanism to be as transparent as possible, we begin by making several simplifying assumptions that we eventually relax. For example, in our baseline model, we adopt a random matching setup with a particular matching function and a simple wagebargaining protocol, while in the online appendix we extend to more general matching functions and show the robustness of our main results to allowing for different bargaining protocols (including directed search). We also focus on the case where the inherited capital produces services that directly enter agents' utility functions. Accordingly, this 7. When excluding the most recent recession, the correlations with depth and length of recovery are, respectively, 0.58 and 0.48, with corresponding p-values 0.1 and 0.17. 8. For our measure of TFP we use Fernald (2014) series which has been corrected for utilization. Note that TFP growth was positive in all recessions except for the two of the early 80s (see Figure 3 in the online appendix). type of capital can be viewed as representing houses or other durable consumer goods. In the online appendix we discuss how the analysis carries over to the case of productive capital.
In our model, trades are decentralized, and there are two imperfections that together cause unemployment risk to be an important factor in household consumption decisions. First, there is a matching friction in the spirit of Diamond-Mortensen-Pissarides, which implies that a household may not be able to find employment. Second, there is limited unemployment insurance, so that households take into account the probability of being unemployed when making consumption decisions. The limited unemployment insurance in our model can be rationalized by invoking a standard adverse selection problem. Since this adverse selection problem can be analyzed separately, in the main body of the text we simply assume that unemployment insurance is not available. 9 The key exogenous variable in the static model will be a stock of consumer durables that households inherit from the past. Our goal is to show why and when high values of this stock can cause the economy to function inefficiently enough to cause a decline in welfare.
The main mechanisms in the model are as follows. Because transactions are not coordinated though centralized markets and unemployment is possible, households view the purchase of durable goods as a risky endeavor, since the ease with which one can pay for these goods will depend on whether or not one becomes unemployed. In particular, households in the model purchase goods on credit. If the rate of unemployment is low, agents will be more willing to purchase durable goods since they know that their future labor earnings are likely to be sufficient to cover the cost of these purchases. In contrast, when unemployment is high, agents tend to hold back on making new purchases, since if they become unemployed they will find it difficult (i.e., costly) to service their resulting debts. Unemployment risk therefore causes consumption decisions across households act as "strategic complements": 10 when one household consumes less, this reduces the demand for goods and thereby increases unemployment, which leads other households to also consume less. The result is a potentially inefficiently large reaction to any impetus.
Inheriting from the past a large amount of durable goods is one such impetus. Because of the above mechanism, the economy can respond perversely to inheriting too many durable goods. In particular, if agents inherit a high level of consumer capital (relative to their desired level, as determined for example by the level of technology), then additional consumer capital has a low marginal value to the household; that is, household demand for new durable goods is low. This is the first-round effect. It is important to note that as a result of this first-round effect, households would never be worse off by inheriting more capital. However, because the initial low demand depresses the labor market, households will perceive the expected cost of purchasing new goods as having increased (since they are more likely to end up in the unemployed state where servicing their debt is more costly), and this will lead to a second-round effect whereby agents further reduce their purchases. A negative multiplier process is then set in motion, with the equilibrium outcome being one where agents buy so little, and economic activity is so depressed, that everyone is made worse off. In this case, agents would benefit from a coordinated increased in purchases, which captures the notion that inherited capital causes a situation of deficient demand. As we shall discuss, a social planner would in 9. In the online appendix we discuss the adverse selection problem that rationalizes the missing insurance market, and use the concomitant information structure to formulate the social planner's problem and compare it to the decentralized outcome.
10. We use here terminology that was popularized in macroeconomic analysis by Cooper and John (1988). such a case want to subsidize hiring by firms so as to help restore efficiency and thereby allow the economy to benefit (instead of suffer) from inheriting more capital.

Setup
Consider an environment populated by a mass L of households indexed by j. In this economy there are two sub-periods, morning and afternoon. In the morning, households may buy a durable good, 11 and try to find employment in the durable-good sector. There will also be a good produced in the afternoon, which we refer to as a service. For now, it is useful to interpret the afternoon period as a reduced-form way of capturing the future. As there is no money in this economy, when the household buys the morning good its bank account is debited, and when (and if) it receives employment income its bank account is credited. Then, in the afternoon, households balance their books by repaying any outstanding debts or receiving a payment for any surplus. These payments are made in terms of the afternoon good (the service), which is also the numéraire in this economy.
Most of the action in the model arises in the morning market. Preferences for the morning are represented by where c represents the flow of consumption services derived from the durable good and is the labor supplied by households in the production of durables. The function U (·) is assumed to be increasing in c, strictly concave and satisfy U > 0. 12 The dis-utility of work function ν(·) is assumed to be increasing and convex in , with ν(0) = 0. Households are initially endowed with X j units of durables, which they can either consume or trade. We assume symmetric endowments, so that X j = X ∀j. 13 In the dynamic version of the model, X will represent the stock of durable goods and will be endogenous.
The key assumption of the model is that trade in the morning is subject to a coordination problem because of frictions in the labor market. In the morning, the household splits up responsibilities between two members. The first member, called the buyer, goes to the durable-good market to make purchases. The second member searches for employment opportunities in the labor market. The market for durables functions in a Walrasian fashion, with both buyers and firms that sell goods taking prices as given. The market for labor in the morning is subject to a matching friction, with sellers of labor searching for employers and employers searching for labor. The information assumption is that buyers do not know, when choosing their purchases of durables, whether the worker member of the household has yet secured a match. This assumption is an easy modeling device that introduces the possibility of cautionary behavior, as buyers will worry about unemployment risk when making purchases.
There is a large set of potential firms operating in the morning who can decide to search for workers in view of supplying durables to the market. Each firm can hire one worker and has access to a decreasing-returns-to-scale production function θF ( ), where is the number of hours worked for the firm and θ > 0 is a technology shift factor. Production also requires a fixed cost θΦ in terms of the output good, so that the net production of a firm hiring hours of labor is θ [F ( ) − Φ]. Φ can be thought of as a 11. We refer to this good as "durable" in anticipation of its role in the dynamic version of the model.
12. We will also assume when needed that the Inada conditions lim c→0 U = ∞ and limc→∞ U = 0 hold.
13. In what follows, we will drop the j index except where doing so may cause confusion.
vacancy-posting cost, as it is incurred before a search can be conducted. For now, we will normalize θ to one, and will reintroduce θ in its general form when we want to talk about the effects of technological change and balanced growth. We will also assume throughout that F (0) = 0 and that Ω( ) ≡ F ( ) is strictly increasing in . 14 Moreover, we will assume that Φ is sufficiently small such that there exists an > 0 satisfying F ( ) − F ( ) = Φ. These restrictions on the production technology are always satisfied if, for example, F ( ) = A α , with 0 < α < 1. We will assume that search is conducted in a random fashion. 15 Given the random search setup, when a firm and a worker match, they need to jointly decide on the number of hours to work and on the wage to be paid. There are many ways the surplus from the match can be divided, as long as it remains within the bargaining set. For the greatest clarity of results, we begin by following Lucas and Prescott (1974) and Guerrieri and Lorenzoni (2009) in assuming that the determination of the wage and of hours worked is done though a competitive pricing process. In effect, one can view a Walrasian auctioneer as calling out a wage w (in units of the afternoon good) that equilibrates the demand for and supply of labor among the two parties in a match. Given that wage, the demand for labor from the firm is therefore given by the marginal productivity condition where p is the relative price of the morning good in terms of the afternoon one. 16 The supply of labor is chosen optimally by the household in a manner to be derived shortly. This competitive pricing process has the feature of limiting any within-pair distortions that could muddle the understanding of the main mechanisms of the model. In the online appendix, we show how our results extend to the case where wages and hours worked are instead determined by a Nash bargaining process. As we show, Nash bargaining introduces additional (and somewhat unintuitive) elements into the analysis that are most easily understood after this baseline framework is presented.
Letting N represent the number of firms who decide to search for workers, the number of matches is then given by the constant-returns-to-scale matching function M (N, L), with M (N, L) ≤ min{N, L}. The equilibrium condition for the durables market is given by where the left-hand side is total purchases of new durables and the right-hand side is the total available supply after subtracting search costs.
Firms will enter the market up to the point where expected profits are zero. The zero-profit condition can be written 17 At the end of the morning, household j's net financial asset position a j , expressed in units of the afternoon good, is given by w j − p(c j − X), with j = 0 if the household working member has not found a job. We model the afternoon so that it is costly to 14. Because we assume free-entry for morning firms, the quantity θΩ( ) will equal net output of durables (after subtracting firms' fixed costs) by a single employed worker.
15. In the online appendix we extend our results to the case of directed search. 16. As will become clear, p can be given an interpretation as a gross interest rate between morning and afternoon.
17. We assume that searching firms pool their ex-post profits and losses so that they make exactly zero profits in equilibrium, regardless of whether they match. arrive in that sub-period with debt. For now, we can simply denote the value of entering the afternoon with assets a j by V (a j ), where we assume that V (·) is increasing and weakly concave, with V (a 1 ) > V (a 2 ) whenever a 1 < 0 < a 2 ; that is, we assume that the marginal value of a unit of assets is greater if one is in debt than if one is in a creditor position. As will become clear, this property of V is precisely the one necessary to generate a cautionary response to changes in unemployment risk, which (as noted earlier) is a fundamental part of the main mechanism of the model. In the following subsection we specify preferences and a market structure for the afternoon that rationalizes a V (·) function of this form.
Taking the function V (a) as given, we can specify the household's morning consumption decision as well as his morning labor-supply decision conditional on a match. The buyer's problem in household j is given by where µ is the probability that a worker finds a job and is given by µ ≡ M (N, L)/L. From this expression, we can see that the consumption decision is made in the presence of unemployment risk. Note that the equilibrium unemployment rate is given by 1 − µ. Conditional on being matched, the worker's problem in household j, taking w as given, can be expressed as choosing a level of hours to supply in the morning so as to solve The first-order conditions associated with these problems are given, respectively, by 18 From these two equations, we can derive household j's consumption demand as a function of the employment rate µ, the relative prices w and p, and the endowment X. Denoting this function c D (µ, w, p, X), it can be verified that 0 ≤ ∂c D (µ,w,p,X) ∂X < 1 and ∂c D (µ,w,p,X) ∂µ ≥ 0. In words, (i ) j's consumption demand is non-decreasing in X, but its demand for new purchases (c D − X) is strictly decreasing in X; and (ii ) j's consumption demand is increasing in the employment rate µ. From this partial-equilibrium perspective (i.e., if µ ,w, and p were held constant), an increase in X would lead to fewer new purchases but (weakly) greater consumption. Moreover, it can be easily verified that the partial-equilibrium effect on welfare of inheriting from the past a larger stock of X can only be positive. Thus, for an increase in X to lead to a fall in consumption and a fall in welfare, it will need to come about through general-equilibrium effects. In our setup, the relevant general-equilibrium effect will run though the employment rate µ: the increase in X will lead to fewer new purchases, which in turn will decrease employment and, since ∂c D (µ,w,p,X) ∂µ > 0, this has the potential to reverse the partial-equilibrium effects and lead to a fall in c and a fall in welfare.
18. As can be seen from the worker's first-order condition, the equilibrium wage w will in general be affected by household j's consumption decision c j . We assume for simplicity that shoppers take the wage as given when making their consumption decision (i.e., they do not try to manipulate the bargaining process through their choice of consumption). Under our later assumptions about V , however, this will not be restrictive.

Deriving the Value Function
V (a j ) represents the value of entering the afternoon with a net financial asset position a j . In this subsection, we derive the function V (possessing the key properties assumed in the previous subsection) by specifying primitives in terms of preferences, technology and market organization. We choose to model the afternoon in such a way that if there were no frictions in the morning there would be no trade between agents in the afternoon. Afternoon preferences are given by where c j is consumption of afternoon services, U (·) is increasing and strictly concave in c j , j is the labor used to produce services, and ν(·) is increasing and convex in j .
To ensure that a unit of net assets is more valuable when in debt than when in surplus, let us assume that households in the afternoon can produce services for their own consumption using one unit of labor to produce θ units of services. However, if a household in the afternoon has to produce market services -that is, services that can be sold to others in order to satisfy debt -then to produce θ units of market services requires them to supply 1+τ units of labor, τ > 0. To simplify notation, we can set θ = 1 for now and return to the more general formulation when talking about the effects of technological change. The continuation value function V (a j ) can accordingly be defined as It is easy to verify that V (a j ) is increasing and weakly concave, with a kink at zero. If ν( j ) is strictly convex, then V (a j ) will be strictly concave, with the key property that V (a 1 ) > V (a 2 ) if a 1 < 0 < a 2 ; that is, the marginal value of an increase in assets is greater if one is in debt than if one is in surplus. 19 Alternatively, if ν( j ) is linear, then V (a j ) will be piecewise linear. Nonetheless, it will maintain the key property that V (a 1 ) > V (a 2 ) if a 1 < 0 < a 2 . We will initially work with this latter case, and in particular will assume that ν( j ) = v j for a constant v. This formulation will greatly reduce the number of general-equilibrium interactions, allowing us to emphasize the channels we think are most relevant. In the online appendix, we discuss how the results are modified when we allow the function V (·) to be a general concave function.

Equilibrium in the Morning Period
Given the function V (·), a symmetric equilibrium for the morning is represented by five objects: two relative prices (the price of durables p and the wage rate w), two quantities (consumption of durables by each household c and the amount worked in each match ), and a number N of active firms, such that 1 c solves the buyer's problem taking µ, p, w and as given; 19. To avoid backward-bending supply curves, we will also assume that ν(·) and U (·) are such that V (a j ) ≥ 0. This assumption is sufficient but not necessary for later results. Note that a sufficient condition for V (a j ) ≥ 0 is that both U (·) ≥ 0 and ν (·) < 0.
2 the labor supply solves the worker's problem conditional on a match, taking p, w and c as given; 3 the demand for labor maximizes the firm's profits given a match, taking p and w as given; 4 the durable goods market clears; 5 firms' entry decisions ensure zero profits.
The morning equilibrium can therefore be represented by the following system of five equations: In the above system, 20 equations (3.5) and (3.6) represent the first-order conditions for the household's choice of consumption and supply of labor. Equations (3.4) and (3.7) represent a firm's entry decision and its labor demand condition. Finally, (3.3) is the goods market clearing condition. At this level of generality it is difficult to derive many results. Nonetheless, we can combine (3.5), (3.6) and (3.7) to obtain the following expression regarding one characteristic of the equilibrium, From equation (3.8), we see that as long as µ < 1, the marginal rate of substitution between leisure and consumption will not be equal to the marginal productivity of work; that is, the economy will exhibit a labor market wedge given by Note that both unemployment risk (µ < 1) and imperfect insurance ( are needed for the wedge to be non-zero. In this environment, contemplating the possibility of being unemployed agents hold back on purchases, which in turn causes the marginal rate of substitution between leisure and consumption to be low relative to the marginal productivity of labor. As we will see, changes in X will cause this wedge to vary, which will cause a feedback effect on economic activity. Our main goal now is to explore the effects of changes in X on equilibrium outcomes. In particular, we are interested in clarifying why and when an increase in X can actually lead to a reduction in consumption and/or welfare. The reason we are interested in this comparative static is that we are interested in knowing why periods of liquidation -that 20. To ensure that an employed worker's optimal choice of labor is strictly positive, we assume is, periods where agents have inherited high levels of durable goods from the past -may be socially painful. To clarify the analysis, we make two simplifying assumptions. First, we assume that the matching function takes the form M (N, L) = Λ min{N, L}, with 0 < Λ ≤ 1. The attractive feature of this matching function is that it has two regimes: one where congestion externalities are concentrated on workers, and one where the externalities are concentrated on firms. In the case where N > L, which we refer to as a tight labor market, workers are the more scarce factor and only the firm matching rate is affected by changes in the ratio N/L. In the case where L > N , which we refer to as a slack labor market, jobs are scarce and only the worker matching rate is affected by changes in the ratio N/L. While this particular matching function is not necessary for our main results, it allows us to cleanly compare the case where congestion externalities are stronger for firms to the case where they are stronger for workers. We will also assume that V (a) is piece-wise linear, with V (a) = va if a ≥ 0 and V (a) = (1+τ )va if a < 0, where τ > 0 and v > 0. This form of the V (·) function corresponds to the case discussed in Section 3.2 where the dis-utility of work in the afternoon is linear. The important element here is τ . In effect, 1 + τ represents the ratio of the marginal value of an extra unit of assets when one is in debt relative to its value when one is in surplus. A value of τ > 0 can be justified in many ways, one of which is presented in Section 3.2. Alternatively, τ > 0 could reflect a financial friction that produces a wedge between borrowing and saving rates. In what follows, we refer to τ as the "marginal cost of debt".
Under these two functional-form assumptions, the equilibrium conditions can be reduced to the following: (3.13) This system of equations now has the feature of being block-recursive, in that equations (3.9), (3.10) and (3.11) can be solved for c, and N , with equations (3.12) and (3.13) then providing the wage and the price. From equations (3.9) and (3.11), one may clearly see the complementarity that can arise between consumption and employment in the case where N < L (i.e., where the labor market is slack). From (3.11) we see that, if N < L, agents will tend to increase their consumption if they believe there are many firms looking for workers (N expected to be large). Then from equation (3.9) we see that, to meet anticipated demand, more firms will need to enter if they believe that consumption will be high, which results in more hiring. Thus, greater consumption favors greater employment, which in turn reinforces consumption. This feedback effect arises as the result of consumption and employment playing the role of strategic complements. Workers demand higher consumption when they believe that many firms are searching to hire, as they view a high N as reducing their probability of entering the afternoon in debt. It is important to note that this complementarity argument is implicitly taking , the number of hours worked by agents, as fixed. But, in the case where the economy is characterized by a slack labor market, this is precisely the right equilibrium conjecture. From (3.10) we can see that if the labor market is slack, then is simply given by , the solution to the equation Λ[F ( ) − F ( ) ] = Φ, and is therefore locally independent of X or c. Hence, in the slack labor market regime, consumption and firm hiring will act as strategic complements. As is common in the case of strategic complements, multiple equilibria can arise. This possibility is stated in Proposition 1. 21 Proposition 1. There exists a marginal cost of debtτ > 0 22 such that (i) if τ <τ , then there exists a unique equilibrium for any value of the durables stock X; and (ii) if τ >τ , then there exists a range of X for which there are multiple equilibria.
While situations with multiple equilibria may be interesting, in this paper we will focus on cases where the equilibrium is unique. Accordingly, Proposition 1 tells us that our setup will have a unique equilibrium if the marginal cost of debt is not too large. For the remainder of this section, we will assume that τ <τ . Proposition 2 focuses on this case and provides a first step in the characterization of the equilibrium.
Proposition 2. When τ <τ , there exists an X such that if X ≤ X then the equilibrium is characterized by a tight labor market (N ≥ L), while if X > X it is characterized by a slack labor market (N < L). Furthermore, there exists an X > X such that if X > X , then employment is zero and agents simply consume their endowment (i.e., c = X). 23 The content of Proposition 2 is very intuitive as it simply reflects that when households have a low endowment of the durable good, they have a high marginal utility. This high marginal utility leads them to purchase many new goods. The high demand is met by new firms entering the market, which leads to a tight labor market. In contrast, if the endowment is high, marginal utility will be low which will reduce the demand for new goods, thereby creating a slack labor market. In this case, variation in demand induces variation in entry decisions, and therefore unemployment, because the free-entry condition acts to create a supply curve for goods that is perfectly elastic. Finally, if X is extremely high, all trade among agents will stop, as people are content to simply consume their endowment. Proposition 3 complements Proposition 2 by indicating how consumption is determined in each regime.
Proposition 3. When the labor market is slack (X > X > X ), the level of consumption is given as the unique solution to When the labor market is tight (X ≤ X ), consumption is the unique solution to 21. The proofs of all propositions are presented in Appendix B.
Finally, when X ≥ X , consumption is given by c = X.
Given the above propositions, we are now in a position to examine an issue of main interest, which is how an increase in X affects consumption. In particular, we want to ask whether an increase in X, which acts as an increase in the available supply of goods, can lead to a decrease in the actual consumption of goods. Proposition 4 addresses this issue.
Proposition 4. If X > X > X , then consumption c is decreasing in the durables stock X. If X ≤ X or X > X , then c is increasing in X.
The content of Proposition 4 is illustrated in Figure 2. Proposition 4 indicates that, starting at X = 0, consumption will continuously increase in X as long as X is compatible with a tight labor market. Then, when X is greater than X , the economy enters the slack labor market regime and consumption starts to decrease as X is increased. Finally, beyond X trade collapses and consumption becomes equal to X and hence it increases with X.
The key result is the existence of a region where total consumption decreases as X increases. The reason why consumption decreases with a higher supply of X in the slack region is because consumption decisions across households play the role of strategic complements in this region. Note that, under our current functional form assumptions, an increase in X leads in partial equilibrium to a one-for-one fall in expenditures, where we define expenditures as e = c − X. If there were no further general-equilibrium effects, the net effect on consumption would be zero. However, there are general-equilibrium effects, and these change depending on whether the labor market is tight or slack. When the labor market is slack, the partial-equilibrium decrease in expenditures reduces the demand for goods as perceived by firms. Less firms then search for workers, which increases the risk of unemployment. The increase in unemployment risk leads households to cut their expenditures further, which further amplifies the initial effect of an increase in X on expenditures. It is because of this multiplier process that an increase in the supply of the good leads to decrease in its total consumption (X + e). Note that such a negative effect does not happen when the labor market is tight, as a decrease in e does not cause an increase in incentives to save, which is the key mechanism at play causing consumption to fall. It is relevant to note that when the labor market is slack, both w and p are invariant to changes in X, and hence there are no general equilibrium effects working though prices in this regime. In contrast, when the labor market is tight, an increase in X leads to a fall in prices which tends to favor an increase in consumption. In this sense, the consumption decisions of households play the role of strategic substitutes when the labor market is tight, so that increased consumption by one households tends to increase prices which in turn causes other households to decrease their consumption.

Figure 2
The link noted above between household j's expenditure, which we can denote by e j = c j − X j , and expenditures by other agents in the economy, which we denote by e, can be captured by rewriting the relations determining e j implied by the elements of Proposition 3 as e j = Z(e) − X, (3.14) with Z(e) ≡ U −1 (Q(e)) (3.15) and ( 3.16) Here, e ≡ Ω( ) is the level of output (net of firms' search costs) that would be produced if all workers were employed, with hours per employed worker equal to . In equilibrium we have the additional requirement that e j = e for all j.
The equilibrium determination of e is illustrated in Figure 3. In the Figure, we plot the function e j = Z(e) − X for two values of X: a first value of X that places the equilibrium in the slack labor market regime, and a second value of X that places it in a tight labor market regime. An equilibrium in this figure corresponds to the point where the function e j = Z(e) − X crosses the 45 • line. Note that changes in X simply move the e j = Z(e) − X curve vertically.

Figure 3
There are several features to note about Figure 3. First, in the case where X ∈ (X , X ), so that the equilibrium of the economy corresponds to a slack labor market with positive trade (i.e., 0 < e < Λe ), the diagram is similar to a Keynesian cross. We can see graphically how an increase in X by one unit shifts down the Z(e) − X curve and, since the slope of Z(e) − X is positive and less than one, a multiplier process kicks in that causes e to fall by more than one. Because of this multiplier process, total consumption of durables, which is equal to e + X, decreases, which is the essence of the first part of Proposition 4. Second, when X < X , so that the labor market is tight (i.e., the equilibrium is such that e > Λe ), the diagram is different from the Keynesian cross. The most notable difference is the negative slope of the function Z(e) − X for values of e > e . This reflects the fact that unemployment risk is not increasing in this regime. In fact, when X is sufficiently small so that the labor market is tight, an increase in X by one unit leads to a decrease in e that is less than one, compared to a decrease of greater than one as exhibited in the slack regime. Here, expenditure by others plays the role of a strategic substitute with one's own expenditure -as opposed to playing the role of a strategic complement as is the case in the unemployment regime -through its effects on real wages and prices. Accordingly, in this region, an increase in X leads to an increase in total consumption of durables. Another more subtle difference with the Keynesian cross is in how the intercept of Z(e) − X is determined. The intercept is given by The X term in the intercept can be interpreted as capturing a pure aggregate-demand effect, whereby higher values of X reduce aggregate demand. However, the remaining term, U −1 ( ν ( ) F ( ) (1 + τ )), reflects technology and preferences. In particular, we can generalize this term by re-introducing the technology parameter θ, in which case the intercept becomes U −1 ( ν ( ) θF ( ) (1 + τ )). 24 In this case, we see that an improvement in technology shifts up the intercept, and will lead to an increase in expenditures. This feature of the Z(e) − X curve illustrates its equilibrium nature, which incorporates both demand and supply effects, as opposed to a Keynesian cross that only reflects demand effects.

Is There Deficient Demand When the Labor Market is Slack?
In the case where X is large enough for the economy to be in the slack labor market regime (X < X < X ), we would like to establish whether this regime should be characterized as suffering from deficient aggregate demand. For this, we need to first define the concept of deficient demand. In our definition we want to focus on a situation where economic activity is inefficiently low and where that low level of activity can be traced back to a lack of demand by others. In particular, we want our definition to exclude a situation where economic activity is inefficiently low simply because of price distortions that are unrelated to a lack of demand by others. For this reason, we define deficient demand as follows.
Definition 1. Deficient demand is a situation where increased demand by one agent would favor increased demand by other agents, and where a coordinated increased in demand by all agents would leave everyone better off.
Using this definition, Proposition 5 indicates that the slack labor market regime is in fact characterized by deficient demand, while the full employment regime is not.
Proposition 5. When the labor market is slack (X < X < X ) it exhibits deficient demand for all τ > 0, while if the labor market is tight (X < X ) the economy does not exhibit deficient demand.

Effects of Changes in X on Welfare
We have shown that when X is high enough, then the labor market will be slack and a local increase in X will cause consumption to fall. We now want to ask how expected welfare is affected in this case, where expected welfare is defined as In particular, we want to ask whether welfare can decrease when the economy is endowed with more goods. Proposition 6 answers this question in the affirmative. Proposition 6 actually goes a step further and indicates two sufficient conditions for there to exist a range of X in the slack labor market regime where an increase in X leads to a fall in welfare.
Proposition 6. An increase in X can lead to a fall in expected welfare. In particular, if either (i) τ is close enough toτ or (ii) the average cost of work ν( ) is low enough relative to the marginal cost of work ν ( ), then there is always a range of X ∈ [X , X ] such that an increase in X leads to a decrease in expected welfare.
Proposition 6 provides an answer to whether more goods can make everyone worse off. In effect, the proposition indicates that the economy can function in a very perverse fashion when households have inherited many goods. We saw from Proposition 4 that an increase in X always leads to a decrease in consumption when the economy is in the slack regime. In comparison, Proposition 6 is weaker as it only indicates the possibility of a fall in welfare in the slack region when X rises. In response to a rise in X in the slack regime, there are three distinct channels through which expected welfare is affected. First, as discussed above, consumption falls, which tends to directly decrease welfare. Second, this fall in consumption is associated with a fall in the probability of being employed. It can be verified that the net benefit of being employed is strictly positive, so that this second effect also tends to decrease welfare. Finally, a rise in X means that a given quantity of consumption can be obtained with a lower level of expenditure, which increases assets for the employed and decreases debt for the unemployed, and therefore tends to increase welfare. Whether this final effect is outweighed by the first two depends on the factors discussed in Proposition 6.

Adding a Non-Durable Goods Sector in the Morning Period
In the previous section we showed that when trade in goods and labor is not simultaneous and there is risk of unemployment, the economy can function in a rather perverse fashion. In particular, we showed that inheriting a large amount of goods can result in deficient demand, with increases in inherited goods reducing both consumption and welfare. In this section we want to briefly explore the robustness of these results to allowing for a second sector in the morning that can potentially expand when the market for durables contracts. This exploration is especially relevant given a common criticism of the liquidation view of recessions, as expressed for example by Krugman (1998), that suggests that liquidationist models cannot explain why we see falls in the consumption of both durable and nondurable goods during most recessions. In order to explore this issue, let us extend the previous setup by making a few small changes. First, let us allow morning utility to take the form U d (c d ) + U n (c n ) − ν( ), where c d is durable consumption and is equal as before to X + e, c n is the added morning non-durable consumption good, and is hours worked. We will assume that there is only one labor market, so that workers can switch frictionlessly across sectors of production. Both durable goods producers and nondurable goods producers will search for workers in this labor market. The markets for durable goods and non-durable goods are assumed to be distinct, with each functioning in a Walrasian fashion. We treat producers of the different goods symmetrically, with production functions in the respective sectors denoted by F d ( ) and F n ( ) (maintaining the assumptions that F s ( ) is increasing in in both sectors s ∈ {d, n}), and assuming that both types of firms face the fixed cost of entering the market given by Φ. Otherwise, we maintain the same structure as before, including the functional-form assumptions for M (N, L) and V (·). The issue we want to examine is the relationship between X and equilibrium outcomes for this economy. Proposition 7 establishes this.
Proposition 7. In the economy with both a durable good and a non-durable good in the morning, if τ is not too large, then for any value of X > 0 there exists a unique equilibrium. Moreover, if Φ is not too small, there exists an X and an X > X , such that: The most interesting aspect of this proposition from our point of view is the existence of a slack labor market regime (when X ∈ [X , X ]), where the consumption of durables, purchases of new durables, and purchases of non-durables all decrease in response to an increase in X. Although workers can be hired by non-durable goods firms in response to an increase in X, the proposition tells us that this substitution does not happen when the labor market is slack. To understand why this does not arise, it is helpful to examine the equilibrium conditions in the case where the labor market is slack, which can be combined to obtain where d and n (hours worked in the unemployment regime) are defined implicitly by the conditions Λ From equations (3.17) and (3.17), we see that when the labor market is slack consumption in the two sectors act as strategic complements, and therefore they move in the same direction in response to a change in X. As long as X is high enough to push the labor market into slack, any further increase in X decreases employment in the durable goods sector, which increases overall unemployment risk. Since this initial increase in unemployment risk causes households to hold back on all of their purchases, demand for both durable and non-durable goods falls, further increasing unemployment risk, and so on. Hence, this version of the model offers an explanation for why inheriting a high level of durables can lead to low activity in both the durable and non-durable goods sectors. 25

Adding a Collateral Constraint
In the previous sections we have illustrated how high levels of inherited durable goods can depress the economy and create a situation of deficient demand. The effects emphasized in the model run in the opposite direction of those emphasized in much of the literature on collateral constraints, wherein higher levels of capital may relax such constraints and favor expenditure by households. In this section we briefly explore how adding a collateral constraint to our model changes the characterization of equilibrium outcomes.
Recall that households in our model were allowed to debit their bank account in order to buy goods in the morning market without any limitations. Now suppose we return to our baseline model (without non-durable goods in the morning) and change it along the several dimensions. First, let us assume that inherited capital goods X also give utility in the afternoon period. 26 Second, we allow creditors to seize capital if debt is not repaid in the afternoon. Third, we assume that a fraction τ > τ of capital is lost when 25. A key condition for the unemployment regime to emerge with high X is that Φ not be too small. Specifically, we require that the value of c n that solves This property is guaranteed if Φ is large enough. 26. Utility in the afternoon is now given by U (X + c) − v , where c is the amount of the afternoon good that is consumed by the household and is the labor used to produce services.
transferring it to a creditor in payment of debt. 27 In this case, a collateral constraint of the form (1 + τ )pe ≤ X will guarantee that households honor their obligations. How does the presence of such a constraint affect how a change in X impacts economic activity? Proposition 8 provides an answer to this question.
Proposition 8. In the presence of a collateral constraint of the form (1 + τ )pe ≤ X, there exist X + and X ++ satisfying 0 < X + ≤ X ++ < X such that equilibrium outcomes are characterized by (i) If X ++ < X < X , then the labor market is slack, there is deficient demand and ∂c ∂X < 0.
(ii) If 0 < X < X + , then the labor market is slack and ∂c ∂X > 1.
Proposition 8 indicates that the presence of a collateral constraint causes the effect of X on c to become richer when compared to the baseline model. In particular, if X is not too large (0 < X < X + ), then, despite high desired consumption, the economy will now exhibit unemployment because the binding collateral constraint limits the amount that households can actually purchase. In this situation, consumption is highly positively responsive to an increase in X, since this relaxes the credit constraint. There is not a coordination problem in this regime, since an increase in the demand of others has no effect on a household's collateral constraint. This regime did not arise in our baseline model. In contrast, when X is high enough (X ++ < X < X ), we once again find that more X leads to less consumption because of the multiplier process discussed previously. In this region, even though agents have enough collateral to spend more, their purchases are still inefficiently low since they do not internalize the effect that their purchasing decisions have on others. It is interesting to note that we could have X + = X ++ , in which case the labor market would always be slack, but the reasons for high unemployment would be very different depending on whether X were high or low. If it were low, unemployment would be high because agents would be limited in how much they can borrow in order to purchase goods that would create employment. In contrast, if X were high, unemployment would be high not because of limited access to credit, but because of the low initial impetus to go out and buy new goods.

DYNAMICS
In this section we explore a dynamic extension of our static model in which morning consumption contributes to the accumulation of X. In particular, we want to consider the case where the evolution of X obeys the accumulation equation where the parameter γ represents the fraction of morning consumption expenditures, e t = c t − X t , that enters the stock of durable goods. Since we do not want to allow heterogeneity between individuals to expand over time, we allow individuals to borrow and lend only within a period but not across periods; in other words, households are allowed to spend more than their income in the morning, but must repay any resulting debt in the afternoon of the same period. The problem facing a household in the morning 27. The assumption that τ > τ guarantees that agents will pay back their debts by working more instead of defaulting. of a period is therefore to choose how many durables to buy and, conditional on a match, how much labor to supply. We model the afternoon of a period as in Section 3.2, where households use labor to produce services either for their own consumption or, at a level of productivity that is lower by a factor 1 + τ , for the consumption of others. In each afternoon, then, the household chooses how many services to consume and to produce to both satisfy its needs and to pay back any accumulated debt. In order to keep the model very tractable, we will continue to assume that dis-utility of work in the afternoon of a period is linear (i.e., equal to v ). Under this assumption, all households will choose the same level of consumption of services in each afternoon, while the production of services will vary across households depending on whether they enter the afternoon in debt or in surplus. Since there are no interesting equilibrium interactions in afternoons, we can maintain most of our focus on equilibrium outcomes in the sequence of morning periods.
Relative to the static case, the only difference in equilibrium relationships (aside from the addition of the accumulation equation (4.17)) is that the first-order condition associated with the household's choice of consumption is now given by the Euler equation ( 4.18) where Q is as defined in equation (3.16). In this dynamic setting, an equilibrium will be represented as a sequence of the previous equilibrium conditions (3.9), (3.10), (3.12) and (3.13), plus the accumulation equation (4.17) and the Euler equation (4.18).
There are many complications that arise in the dynamic version of this model, which makes characterizing equilibrium behavior more difficult. In particular, there can be multiple equilibrium paths and multiple steady-state solutions. Luckily, the problem can be simplified if we focus on cases where δ is small; that is, on cases where the durability of goods is high. In addition to simplifying the analysis, focusing on the low-δ case appears reasonable to us, as many consumer durables are long-lived, especially if we include housing in that category. In the case where δ is sufficiently small, as stated in Proposition 9, the economy will have only one steady state and that steady state will have the property that the labor market will be slack.
Proposition 9. If δ is sufficiently small, then the model has a unique steady state and this steady state is characterized by a slack labor market.
Proposition 9 is very useful, as it will allow us to analyze the equilibrium behavior around the unique steady state. Accordingly, for the remainder of this section, we will assume that δ is sufficiently small so that Proposition 9 applies.

Local Dynamics
In this subsection, we explore the local dynamics assuming that δ is small enough that the steady state is unique and in the slack labor market regime. The first question we address is whether equilibrium dynamics can exhibit local indeterminacy. In other words, can forward-looking behavior give rise to an additional potential local source of multiple equilibria in our setup? Proposition 10 indicates that this is not possible; that is, the roots of the system around the unique steady state can not both be smaller than one. 28 28. In this section we only consider local dynamics around a unique unemployment-regime steady state. Nonetheless, it is straightforward to show that if the unique steady state is in the full-employment regime, then the local dynamics necessarily exhibit monotonic convergence.
Proposition 10. The local dynamics around the steady state can either exhibit monotonic convergence in c and X, convergence with oscillations, or divergence. Locally indeterminacy is not possible.
Proposition 10 is useful as it tells us that the decision rule for consumption around the steady state is a function. 29 Accordingly, we can now examine the sign of the derivative of this function; that is, whether the decision rule for consumption around the steady state has the property that a larger X leads to a lower level of consumption, as was the case in our static model when in the slack labor market regime. Proposition 11 indicates that if τ is not too large, then local dynamics will exhibit this property. Note that the condition on τ is a sufficient condition only.
Proposition 11. If τ is sufficiently small, then in a neighborhood of the unique steady state, consumption is decreasing in X, with the dynamics for X converging monotonically to the steady state.
From Proposition 11 we now know that, as long as τ is not too big, our model has the property that when the economy has over-accumulated relative to the steady state (i.e., if X slightly exceeds its steady-state value), then consumption will be lower than in the steady state throughout the transition period, which we may refer to as a period of liquidation. In this sense, the economy is overreacting to its inherited excess of capital goods during this liquidation period, since it is reducing its expenditures to such an extent that people are consuming less even though there are more goods available to them in the economy. While such a response is not socially optimal, it remains unclear whether it is so excessive as to make people worse off in comparison to the steady state, since they are also working less during the liquidation phase. It turns out that, as in the static case, the welfare effect of such a liquidation period depends, among other things, on whether the average dis-utility of work is small enough relative to the marginal disutility. For example, if the average dis-utility of work is sufficiently low relative to its marginal value, then it can be verified that a liquidation period induced by inheriting an excess of X will make average utility in all periods of the transition lower than the steady-state level. This result depends in addition on the unemployment rate not being too large in the steady state.
While we do not have a simple characterization of the global dynamics of the model, Propositions 10 and 11 suggest that, starting from X = 0, the economy will likely go though a phase with a tight labor market, with both X and c increasing over time. The economy then enters into a range with a slack labor market once X is large enough. Then, as long as τ is not too great, X will continue to monotonically increase, converging toward its steady state. In contrast to X, upon entering the slack regime, consumption starts to decrease as unemployment risk increases and causes household's to hold back on purchases, which further depresses activity. Eventually, the economy will reach a steady state where consumption, employment, and possibly period welfare are below the peak levels reached during the transition.

A Quantitative Model of Booms and Busts with Irrational Exuberance
In this section, we illustrate the possibility of generating boom-bust cycles with the type of mechanism that we have put forward. In particular, we evaluate the model's ability to match the suggestive evidence presented in Section 2. As mentioned earlier, our aim is not to propose a novel explanation as to why the economy may embark on periods of rapid accumulation that turn out to be ex post excessive. Possible explanations could be noisy news, financial shocks (which could be modeled as changes in the marginal cost of debt, τ , in our model), uncertainty, waves of optimism, etc. For the purposes of illustration, in this section we suppose that the economy randomly undergoes episodes of excessive optimism about the growth of the economy, which we call irrational exuberance.
To derive some quantitative implications, we make several functional-form assumptions and then calibrate the model to reproduce salient features of the U.S. postwar business cycle. The model is written in per capita terms, and the length of a period is taken to be a quarter. We assume that the period utility and production functions are given, respectively, by log(c) − ν 1+ω 1+ω and θ t A α , where total factor productivity θ t follows a deterministic trend with growth rate γ θ . On top of the parameters ν, ω, A, γ θ and α, we need to calibrate the discount factor β, the depreciation rate δ, the share of durable goods in total expenditure γ, the vacancy-posting cost Φ, the marginal cost of debt τ , and the number of participating workers L.
We set γ θ = 1.0048 to match average per capita GDP growth and normalize θ 0 to 1. We set the discount factor to the commonly chosen level β = 0.99. We interpret X as the stock of durable goods. Expenditures on durable goods are on average over the postwar period 8.5% of total expenditures, so that we set γ = 0.085. 30 Similarly, the depreciation rate is calibrated to the the average ratio of "Current-Cost Depreciation of Consumer Durable Goods" over the "Current-Cost Net Stock of Consumer Durable Goods", so that δ = 0.045. We set α = 2/3 to obtain a labor income share of 2/3. The Frisch elasticity of labor supply in the intensive margin ω is set to 0.5 following Chetty et al. (2011). A and L are normalized to one. ν and Φ are then set to target the postwar average unemployment rate of 6% and to yield the normalized level of hours per capita = 1.
Episodes of irrational exuberance are modeled in the following way. The economy can be either in normal times or in an episode of exuberance, but actual TFP always grows at factor γ θ . In normal times, actual growth is correctly anticipated by economic agents when making spending decisions. With probability 1 − p 1 , the economy remains in the normal regime in the subsequent period, while with probability p 1 it enters into an episode of irrational exuberance. When the economy is exuberant, agents initially optimistically believe that TFP grew at a higher-than-usual factor κγ θ from last period, where κ > 1 measures the degree of optimism. Shoppers make their purchase decisions according to this optimistic belief. On the other hand, upon entering the labor market workers learn the true level of TFP and receive a wage commensurate with it. Accordingly, households "overspend" and "overborrow" in the morning, thereby "overaccumulating" the durable good X. When in an exuberant period, the economy remains exuberant into the next period with probability p 2 , and returns to normal with probability 1 − p 2 . Given the mechanisms of our model, when the economy returns to normal times it does so with 30. We interpret consumption in the model as reflecting both the services from existing durable goods, plus the felicity from current expenditures, where current expenditures can be on both durable and non-durable goods. For this reason, we set γ < (1 − δ), which implies that agents know that only a fraction γ of these expenditures will become durable goods. a high stock of durable goods X, which is the main force that triggers a recession. 31 The simulated model will thus feature booms and busts, even though actual TFP never deviates from trend.
There are four parameters that are specific to our model and shock structure that cannot be taken from the literature: the marginal cost of debt parameter τ , the level of optimism in irrational exuberance episodes κ, and the transition probabilities p 1 and p 2 . We calibrate these parameters to match four moments of the U.S. business cycle: 32 the standard deviation of output growth (0.96%), the average depth of a recession (-2.9%), the average length of a recovery (6 quarters) and the number of recession episodes per 100 quarters (3.3 episodes) (see Table 2). 33 Using this procedure, the calibrated values of the parameters are as follows. The degree of optimism is κ = 1.22, meaning that in an optimistic quarter, agents believe that the growth rate of TFP is 0.585% instead of the average rate of 0.48%. The marginal cost of debt is given by τ = 0.065. The probability of entering into an optimistic episode is p 1 = 0.3, and the probability of remaining in an exuberant episode is p 2 = 0.48. With this calibration, exuberant episodes last for an average of two quarters, and it takes at least three quarters of exuberance to produce a recession. 34 Table 2 Next, using data simulated from our calibrated model, we perform the same empirical analysis conducted on actual data in Section 2: we compute cumulated investment, and obtain the correlations between it and both the depth of recessions and the length of recoveries. Note that neither of these correlations were targeted in the calibration exercise. Nonetheless, as shown in Table 2, the model does a very good job at matching both of these moments. 35 As an illustration, Figure 4 shows scatter plots of depth and length against cumulated investment for a typical simulation of our model.

Figure 4
To illustrate the mechanism at play in a boom-bust cycle driven by irrational exuberance, Figure 5 displays the path of an economy that is on a balanced growth path until period 4, and then, beginning in period 5, enters into an 8-quarter episode of exuberance. As shown in the upper panel of the figure, GDP, as measured by total expenditure, increases faster and above trend during the exuberance episode, which translates into a corresponding over-accumulation of durable goods (lower panel). When 31. An alternative way to model over-optimistic beliefs would be to assume rational (Bayesian) learning by economic agents in an environment with permanent and transitory shocks to TFP and/or noisy signals, as in, for example Beaudry and Portier (2004), Jaimovich and Rebelo (2009) (2014) for a survey of this literature). Because the reaction of the economy to excessive accumulation in our model is quite independent of the precise mechanism that generates optimistic episodes, we adopt here the analytically simpler approach of irrational exuberance. 32. To obtain moments for the model we average the relevant statistics from 10,000 simulations of the model of 270 quarters each in length.
33. To detect peaks and troughs, we use Harding and Pagan (2002) version of Bry and Boschan (1971) algorithm, as provided by James Engel. Episodes of decreasing output are classified as recessions if total output growth is smaller than -2% in total.
34. Recall that a recession is an episode in which output per capita decreases. Given the secular trend in TFP, under our calibration short periods of exuberance generate negative deviations from trend output per capita, but not a negative growth rate.
35. Note that, since TFP in the model does not fluctuate around its trend, controlling for TFP growth will have no impact, just as we found in the data ( Table 1 in Section 2). agents realize their error in period 12, they cut back on their expenditure to start a liquidation process. 36 Insert Figure 5 5. DYNAMIC POLICY TRADE-OFFS In this final section, we turn to one of our motivating questions and ask whether or not stimulative policies should be used when an economy is going through a liquidation phase characterized by high unemployment. In particular, we consider the case where the economy has inherited from the past a level of X above its steady-state value and, in the absence of intervention, would experience a period of liquidation, with consumption below its steady-state level throughout the transition. Obviously, the first-best policy in this environment would be to remove the sources of frictions or to perfectly insure agents against unemployment risk. However, for reasons such as adverse selection or moral hazard, such first-best policies may not be possible. We therefore consider the value of a more limited type of policy: one that seeks only to temporarily boost expenditures. In particular, we are interested in asking whether welfare would be increased by stimulating expenditures for one period, knowing that this would imply a higher X (and therefore lower consumption) in all subsequent periods, thereby prolonging the recession. This question is aimed at capturing the tension between the Keynesian and Hayekian policy prescriptions discussed earlier. 37 In order to understand the policy we are considering, recall that the law of motion for X is given by where the function e(X t ) is the equilibrium policy function for e t . Now, beginning from steady state, suppose at t = 0 we stimulate expenditures by for one period such that the stock at t = 1 is now given by As as result of this one-period stimulus, the path of expenditures for all subsequent periods will be changed even if there is no further policy intervention. The new sequence for X, which we denote X t , will be given by X t+1 = (1 − δ) X t + γe( X t ) for all t ≥ 1. We now examine the effect on welfare of such a policy. To do so, we focus on the case where the steady state of the system is stable and δ is sufficiently small that the labor market is slack at the steady state. In this case, we have the following result.
Proposition 12. Suppose the economy is undergoing a liquidation (i.e., X, and therefore the unemployment rate, are above the steady state). Then a small temporary 36. In the online appendix, we study the pattern of fluctuations generated by a simple RBC model subject to irrational exuberance shocks. Periods of optimism are associated with high levels of activity but low investment, so that cumulated investment is below trend at the onset of a recession, and its correlation with depth of recessions and length of recoveries is negative. We also model the case of TFP shocks, and show that those correlations are small and insignificant.
37. In answering this question, we will be examining the effects of such a policy without needing to be very explicit about the precise policy tools used to engineer the stimulus, which could come from a number of different sources. For example, the stimulus we consider could be engineered by a one-period subsidy to consumption financed by a tax on the employed. stimulus will increase welfare even though it delays the return of unemployment to its steady state level.
Proposition 12 indicates that when the economy is experiencing a liquidation, a oneperiod policy of stimulating household expenditures can increase welfare even though it postpones the recovery. This arises because the gain in utility coming from the initial stimulus is greater than the loss in utility associated with a more prolonged recession. With respect to the policy debate between the followers of Hayek and Keynes, we take the results from Proposition 12 as indicating that Hayek may have been right to emphasize the cost of stimulus policies in terms of prolonging a recession, but it may nevertheless still be desirable to stimulate demand in such a case -at least by a small amount-since in the absence of intervention the market outcome will tend to generate a recession that is excessively deep.

CONCLUSION
There are three elements that motivated us to write this paper. First, there is the observation that many recessions arise after periods of fast accumulation of capital goods, either in the form of houses, consumer durables, or productive capital. This, in our view, gives plausibility to the hypothesis that recessions may often reflect periods of liquidation where the economy is trying to deplete excesses from past over-accumulation. 38,39 Second, during these apparent liquidation-driven recessions, the process of adjustment seems to be socially painful and excessive, in the sense that the level of unemployment does not seem to be consistent with the idea that the economy is simply "taking a vacation" after excessive past work. Instead, the economy seems to be exhibiting some coordination failure that makes the exploitation of gains from trade between individuals more difficult than in normal times. These two observations capture the tension we believe is often associated with the Hayekian and Keynesian views of recessions. Finally, even when monetary authorities try to counter such recessions by easing policy, this does not seem to be sufficent to eliminate the problem. This leads us to believe that there are likely mechanisms at play beyond those related to nominal rigidities. Hence, our objective in writing this paper was to offer a framework that is consistent with these three observations, and accordingly to provide an environment where the policy trade-offs inherent to the Hayekian and Keynesian views could be discussed.
A central contribution of the paper is to provide a simple macro model that explains why an economy may become particularly inefficient when it inherits an excessive amount of capital goods from the past. The narrative behind the mechanism is quite straightforward. When the economy inherits a high level of capital, this decreases the desire for trade between agents in the economy, leading to less demand. When there are fixed costs associated with employment, this will generally lead to an increase in unemployment. If the risk of unemployment cannot be entirely insured away, households will react to the increased unemployment by limiting purchases and thereby further depressing demand. This multiplier process will cause an excess reaction to the inherited goods and can be large enough to make society worse off even if -in a sense -it is richer 38. Note that this is a fundamentalist view of recessions, in that the main cause of a recession is viewed as an objective fundamental (in this case, the level of capital relative to technology) rather than a sunspot-driven change in beliefs.
39. An alternative interpretation of this observation is that financial imbalances associated with the increase in capital goods are the main source of the subsequent recessions.
since it has inherited a large stock of goods. Within this framework, we have shown that policies aimed at stimulating activity will face an unpleasant trade-off, as the main effect of stimulus will simply be to postpone the adjustment process. Nonetheless, we find that such stimulative policies may remain desirable even if they postpone recovery.

APPENDIX A. DATA
In sections 2 and 4.2, we use a sample that runs from 1947Q1 to 2015Q2. Series definitions are: -Peaks and troughs dates are taken from the NBER website.
-Population : FRED, Total Population: All Ages including Armed Forces Overseas (POP), Thousands, Quarterly, Not Seasonally Adjusted, starts in 1952. For 1948to 1952  Denoting i Xt real investment of quarter t, where X is either consumer durables, fixed, non residential and residential and δ X the depreciation rate, cumulated investment ci Xt between t and t − T + 1 is computed as ci Xt = T −1 j=0 δ j X i Xt−j .

APPENDIX B. PROOFS OF PROPOSITIONS
For simplicity, when the matching function is of the form M (N, L) = Λ min{N, L} we prove results only for the case where Λ = 1. It is straightforward to extend all proofs to the more general case.

Proof of Proposition 1
We first establish that there always exists an equilibrium of this model. Substituting equation (3.9) into equation (3.11) and letting e ≡ c − X yields where Ω( ) ≡ F ( ) , which is assumed to be strictly increasing. When N < L, equation (3.10) implies that = , and equation (3.9) implies that e < e , where e ≡ Ω( ). On the other hand, when N > L, equation (3.9) implies that = Ω −1 (e). Further, since min{N, L} < N and F ( ) − F ( ) is strictly increasing in , equation (3.10) implies that > , and thus, by strict increasingness of Ω, we also have e > e . Substituting these results into equation (B19) yields that e > 0 is an equilibrium of this model if it satisfies where the function Q(e) is defined in equation (3.16). Note that Q is continuous, strictly decreasing on [0, e ], and strictly increasing on [e , ∞).
Proof. For the first part, suppose aggregate conditions are that e = 0. Then the household's marginal utility of consumption from simply consuming its endowment is no greater than its expected marginal cost. Households thus respond to aggregate conditions by making no purchases, which in turn validates e = 0. For the second part, we have that min Q(e) = ν ( )/F ( ) > 0. Since limc→∞ U (c) ≤ 0 by assumption, it follows that, for any X, there exists an e sufficiently large that U (X + e) < min Q(e), and therefore there must exist a solution e > 0 to equation (B20).
Lemma 1 implies that an equilibrium necessarily exists. We now to showing under what conditions this equilibrium is unique for all values of X. As in equation (3.14), let e j (e) = U −1 (Q(e)) − X denote j's optimal expenditure when aggregate expenditure is e, so that equilibrium is a point e j (e) = e. e j (e) is continuous everywhere, and differentiable everywhere except at e = e , with e j (e) = Q (e) U (U −1 (Q(e))) .
Note that e j (e) is independent of X, strictly increasing on [0, e ] and strictly decreasing on [e , ∞). Proof. For e > e , e j (e) < 0 , so that e j (e) < 1 is satisfied. For e < e , note that e j (e) = Q (e) − U (X + e j (e)) e j (e) 2 U (X + e j (e)) .

Lemma 2. If
Since Q (e) = 0 on this range and U > 0, we have e j (e) > 0, and thus e j (e) < lim e↑e e j (e), which completes the proof.

Lemma 3. Inequality (B21) holds if and only if
Proof. We have that , which is clearly less than one if and only if τ <τ .
Lemma 4. If τ <τ , then there always exists a unique equilibrium regardless of the value of X. If τ >τ , then there exists values of X ∈ R such that there are multiple equilibria.
Proof. Note that an equilibrium in (e, e j )-space is a point where the e j = e j (e) locus intersects the e j = e locus. To see the first part of the lemma, suppose τ <τ so that inequality (B21) holds. Then since the slope of the e j = e locus is one, and the slope of the e j = e j (e) locus is strictly less than one by Lemma 2, there can be only one intersection, and therefore the equilibrium is unique.
To see the second part of the lemma, suppose that τ >τ and thus (B21) does not hold. Then by strict convexity of e j (e) on (0, e ), there exists a value e < e such that e j (e) > 1 on (e, e ). Define X(e) ≡ U −1 (Q(e)) − e, and note that e is an equilibrium when X = X(e). We show that there are at least two equilibria when X = X(e) with e ∈ (e, e ). To see this, choose e 0 ∈ (e, e ), and note that, for X = X(e 0 ), e j (e 0 ) = e 0 and e j (e) > 1 on (e 0 , e ). Thus, it must also be the case that e j (e ) > e . But since e j (e) is continuous everywhere and strictly decreasing on e > e , this implies that there exists some value e > e such that e j (e) = e, which would represent an equilibrium. Since e 0 < e is also an equilibrium, there are at least two equilibria.
This completes the proof of Proposition 1.

Proof of Proposition 2
Lemma 5. If τ <τ and X is such that e > 0, then de/dX < 0.
Proof. Differentiating equilibrium condition (B20) with respect to X yields From Lemma 3, Q (e) > U U −1 (Q(e)) . In equilibrium, U −1 (Q(e)) = X + e, so that this inequality becomes Q (e) > U (X + e), and thus the desired conclusion follows by inspection.
Given Lemma 5 and the fact that the economy exhibits unemployment when e < e and full employment when e ≥ e , the economy will exhibit unemployment if and only if X ≤ X , where is the level of X such that e = e is the equilibrium. This establishes the first part of the proposition. Next, from Lemma 1, we see that there is a zero-employment equilibrium if and only if U (X) ≤ ν ( ) F ( ) (1 + τ ), which holds when X ≥ X , where This completes the proof of Proposition 2.

Proof of Proposition 3
If X < X , we know from Proposition 2 that e > 0, and therefore e solves equation (B20). Substituting e = c − X for e yields the desired result in this case. From Proposition 2, we also know that if X ≥ X then e = 0, in which case c = X, which completes the proof.

Proof of Proposition 4
If X > X then e = 0 and c = X, so that c is increasing in X. Suppose instead X < X , so that e > 0. Differentiating c = X + e with respect to X and using equation (B22), we obtain Since the denominator of this expression is positive (see proof of Lemma 5), the sign of dc/dX is given by the sign of Q (e), which is negative if e < e (i.e., if X < X < X ) and positive if e > e (i.e., if X < X ). This completes the proof.

Proof of Proposition 5
Letting U (e) denote welfare conditional on the coordinated level of e, we may obtain that where µ(e) = e/[F ( ) ] denotes employment conditional on e and L ≡ ν ( ) − ν( ) ≥ 0. Using the envelope theorem, the only welfare effects of a marginal change in e from its decentralized equilibrium value are those that occur through the resulting change in employment. Thus, and therefore a coordinated rise in e would increase expected utility of all households.

Proof of Proposition 6
Denote welfare as a function of X by If X < X (full-employment regime) or X > X (zero-employment regime), U (X) > 0 clearly always holds. Thus, we focus on the case where X ∈ (X , X ), in which case some algebra yields Using the envelope theorem, we may differentiate this expression with respect to X to obtain where L ≡ ν ( ) − ν( ) ≥ 0.
Proof. Substituting the equilibrium condition (B20) into (B24) and using the fact that dµ/dX = [F ( ) ] −1 de/dX, after some algebra we obtain From (B22), we may also obtain that and therefore Since de/dX < 0, dµ/dx < 0 and thus the first term is positive, as is the second term, and the proof is complete.
1+τ then there exists a range of X such that U (X) < 0.
Proof. Since U is convex by Lemma 6, U (X) < 0 for some X if and only if lim X↓X U (X) < 0.
Taking limits of equation (B25), and using the facts that lim X↓X de dX = −τ / (τ − τ ) and lim X↓X µ = 1, we obtain that Substituting in from the definition of L , straightforward algebra yields that this expression is less than one if and only if τ > τ .
Note that, by convexity of ν( ) and the fact that ν(0) = 0, we have ν( ) ≤ ν ( ) , and thus τ <τ , so that there always exists values of τ such that τ < τ <τ . From the definition of τ , we also see that, holding τ and ν ( ) constant, if ν( )/ is small, this inequality is more likely to be satisfied.

Proof of Proposition 7
In the extended model with non-durables, the equilibrium conditions for quantities can be written 40 We first consider equilibria within three regimes separately, then establish the relationship between these regimes and different ranges of X. We assume that lim c n →0 U n (c n ) = ∞, so that we will always have N n > 0. In our notation, the superscript s will index the sector, with s ∈ {d, n}. Equilibrium Regime 1: N ≡ N d + N n ≥ L. Suppose N ≡ N d + N n ≥ L, so that µ = 1. Assuming Φ is small enough to ensure the existence of a regime with µ < 1 (see footnote 25), we must have N d > 0 when µ = 1. Letting ψ ≡ N d /N , and letting p s ( ) ≡ ν ( ) /F s ( ), the equilibrium conditions can be written Lemma 8. There exists at most one equilibrium in Regime 1.
Proof. The zero-profit conditions (B26) and (B26) can be solved to obtain s = s (N ), with s (N ) > 0. Substituting the resource constraint equations (B26) and (B26) into the demand equations (B26) and (B26) for e and c n , we may then reduce the system to two equations, U n ((1 − ψ) Ω n ( n (N ))) = p n ( n (N )) , 40. The prices and wages are given by in two unknowns, N and ψ. Each of these equations gives a locus of points in (N, ψ)-space, with intersections representing equilibria. Totally differentiating with respect to N yields, respectively, so that the locus associated with (B26) is downward-sloping, and the locus associated with (B27) is upward-sloping. Thus, if an equilibrium exists in this region, it is clearly unique.
Proof. A rise in X results in a shift down of the locus associated with equation (B26), so that dψ/dX < 0 and dN/dX < 0. Since a fall in N results in a fall in s and thus also p s , it follows from equations (B26) and (B26) that dc d /dX > 0 and dc n /dX > 0. Further, since ψ and d both fall, it follows from equation (B26) that de/dX < 0.
Equilibrium Regime 2: N < L and N d > 0. Suppose now that N ≡ N d + N n < L but with a positive output of durables (N d > 0).
Lemma 10. For τ sufficiently small, there exists at most one equilibrium in Regime 2.

Proof. Let
Z n (e, c n ) ≡ U n −1 ν ( n ) F n ( n ) 1 + τ − τ e Ω d d + c n Ω n ( n ) where, as before, Ω s ( ) ≡ F s ( ) . For a given e, c n solves c n = Z n (e, c n ). This solution will always be unique if Z n 2 (e, c n ) < 1 for all combinations of e and c n such that N ≤ L. Since Z n 2 (e, c n ) = 1 −U (Z n (e, c n )) ν ( n ) F n ( n ) τ Ω n ( n ) , Z n 2 (e, c n ) is maximized when Z n (e, c n ) is maximized, which in turn occurs when µ = 1. Thus, a sufficient condition to ensure that c n is always uniquely determined given e is that Assume henceforth that this is true, and let c n (e) denote the unique value of c n that solves c n = Z n (e, c n ). Note that Since τ <τ n , it may be verified that the denominator of this expression is strictly positive, so that c n > 0. Further, c n (e) = ν ( n ) if e > e Note that p (e) is constant (and equal to p ) for e < e , while p (e) > 0 for e > e . Thus, X/ [(1 + τ ) p (e)] is constant on e < e and decreasing on e > e . LetX 1 ≡ (1 + τ ) p e and letX 0 be the unique solution satisfyingX ≥X 1 to provided such a solution exists.X 0 is the value of X (if it exists) that places the equilibrium in the slack region with a collateral constraint that just binds. It can be verified that the solution to e = U −1 (Q (e)) − X satisfies e < min {e , X/ [(1 + τ ) p ]} forX 0 < X < X 41 whenX 0 exists, and for X < X < X whenX 0 does not exist. Thus, letting X ++ =X 0 ifX 0 exists, and X ++ = X otherwise, we have that, for X ++ < X < X , the equilibrium is in the slack region and features a non-binding collateral constraint. Part (i) of the Proposition then follows immediately.
Next, note that the equilibrium is in the slack region with a binding collateral constraint for X <X 0 ifX 0 exists, and for X <X 1 ifX 0 does not exist. Thus, letting X + =X 0 ifX 0 exists, and X + =X 1 otherwise, we have that, for X < X + , the equilibrium is in the slack region and features a binding collateral constraint. Further, since e = X/ [(1 + τ ) p ] in this case, we have ∂e/∂X > 0, and thus ∂c/∂X > 1, which completes the proof of part (ii) of the Proposition.
Lemma 13. For δ sufficiently small, a steady state exists and is unique.
Proof. Similar to in the static case, we may express individual j's optimal choice of steady-state expenditure e j given aggregate steady-state expenditure e as e j (e) = δ δ+γ U −1 (ζQ(e)). As before, we can verify that e j (e) < 0 for e > e , while e j (e) > 0 and e j (e) > 0 for e < e . Thus, an equilibrium necessarily exists and is unique if e j (e) < 1 for e < e , which is equivalent to the condition that lim e↑e e j (e) < 1. This is in turn equivalent to the condition τ < τ , where As δ → 0, τ approaches infinity, and thus it will hold for any τ , which completes the proof.
41. Note that we must haveX 0 < X . Suppose not, i.e., supposeX 0 ≥ X . Then if X =X 0 agents would not want to make any purchases in equilibrium (e = 0), in which case the collateral constraint e ≤ X/ [(1 + τ ) p * ] would not bind, contradicting the definition ofX 0 .
Lemma 16. The system is saddle-path stable with positive eigenvalues if and only if (1 − δ − γ)U (X + e) < (1 − δ)Q (e). (B35) Proof. Note that the system is stable with positive eigenvalues if and only if (B34) holds. We have Since the numerator is positive by (B30), inequality (B34) holds if and only if (B35) holds.
Lemma 17. If then the system is saddle-path stable with positive eigenvalues.
Proof. Note that condition (B35) always holds around a full-employment steady state. If the steady state is in the unemployment regime, then condition (B35) holds if and only if e j (e) < δ δ+γ ζ 1−δ−γ 1−δ ∈ (0, 1), where e j (e) is as defined in Lemma 13. As before, this condition holds for all e if it holds for lim e↑e e j (e), which it can be verified is equivalent to the condition τ < τ . Note also that τ < τ , where τ was defined in equation (B29), so that this condition is strictly stronger than the one required to ensure the existence of a unique steady state.
Lemmas 16 and 17 together establish that, for τ sufficiently small (e.g., τ < τ ), the system converges monotonicaly to the steady state. It remains to show that consumption is decreasing in the stock of durables. Assuming τ is sufficiently small so that the system is saddle-path stable with positive eigenvalues, it is straightforward to obtain the solution Xt = λ t 1 X 0 , et = ψ Xt, ct = (1 + ψ) Xt, where ψ ≡ −(1 − δ − λ 1 )/γ. Thus, consumption is decreasing in the stock of durables if and only if ψ < −1.
Lemma 18. If (B35) holds and the steady state is in the unemployment regime, then ψ < −1.
Proof. We may write

Proof of Proposition 12
Let et (X 0 , ) and Xt (X 0 , ) denote the paths for expenditure and the stock of durables, respectively, when the initial stock of durables is X 0 and there is stimulus of in period 0. Letting e (Xt) denote the (de-centralied) equilibrium policy function for e, we thus have X 0 (X 0 , ) = X 0 , e 0 (X 0 , ) = e (X 0 ) + , and, for t ≥ 1, Xt (X 0 , ) = (1 − δ) X t−1 (X 0 , ) + γ e t−1 (X 0 , ) and et (X 0 , ) = e Xt (X 0 , ) . Let U (X 0 , ) denote the corresponding welfare as a function of X 0 and . If U (X, 0) > 0 (where X is the steady state stock of durables), then it follows that, for X 0 greater than but sufficiently close to X (so that the economy is undergoing a liquidation), we will have U (X 0 , 0) > 0, i.e., a small one-period stimulus will enhance welfare. We thus turn now to establishing that U (X, 0) > 0. Using the envelope condition, it is straightforward to obtain that U (X, 0) = where e is the steady state level of purchases. Thus, U (X, 0) is positive if the summation in this expression is positive. One may obtain that Since e (X) = − (1 − δ − λ 1 ) /γ, where λ 1 is the smallest eigenvalue of the dynamic system (in modulus), we have that ∂ et (X, 0) /∂ = − (1 − δ − λ 1 ) λ t−1 1 for t ≥ 1. Thus, ∞ t=0 β t ∂ et (X, 0) ∂ = 1 − β (1 − δ) 1 − βλ 1 If the system is saddle-path stable, then |λ 1 | < 1, and thus this expression is strictly positive, which completes the proof.